Hi Daniel - Welcome to the Community!
The Story issue type will follow the path of your workflow attached to that issue type. So, it should eventually end up in a Done status.
Can you verify the workflow it is using? (Click on the View Workflow link below the status in the "old" JIRA Issue view).
It probably has to do with following the Workflow - Check the flow diagram to see the conditions that arrive at 'Done' state.
eg your workflow may recognize only items (tasks/ Stories) in 'Dev Testing' or 'QA' can transition to Done state. ie. such case, you have to go to the Story details and bring the Story status to Dev Testing or QA before you can take the Story to Done state.
If it is cumbersome, redefine the workflow. Hope that Helps
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